Once I heard about the Ulam spiral, I started wondering if there were other ways in which prime numbers could be arranged that result in interesting patterns. A few are already known, including the Klauber triangle and the Sacks spiral. But are there any others?
It’s a good test case for AI. I asked Anthropic’s Fable, which is the world’s most powerful out-of-the-box LLM available to mere mortals like me. The prompt I used is below.
The disappointing answer: Every candidate Fable came up with traced back to known or trivial mechanisms. There are no simple, new ways to arrange prime numbers that reveal unknown patters. This is maybe not surprising, given that many of the world’s best mathematical minds have thought about this for many decades.
Here is my Fable prompt:
There are several variants of the Ulam spiral, including the Klauber triangle and the Sacks spiral. Look for other known variants of marking prime numbers on integers that have been arranged following a simple rule so that patterns emerge. Then try to test additional simple arrangements that haven’t been published, and test each for the emergence of visual patterns like in the Ulam Spiral. Devise a suitable test to identify the presence of visual patterns. Don’t test trivial iterations on known patterns (e.g. counterclockwise spiral). You may first test new arrangements you judge to be most likely to result in patterns. If you find a non-trivially new, previously unpublished arrangement from which a clear pattern emerges, notify me, return a picture of the arrangement, and a short text stating how the arrangement was made and what the pattern is. Limit to the first 1E6 integers (78,498 primes) for testing. If you have tested 200 arrangements and have not found any visual patterns, stop and notify me.
Nehaveigur 